Abstract:
We consider the problem of minimizing the eigenvalues of the Schrödinger operator H=−Δ+αF(κ) (α>0) on a compact n-manifold subject to the restriction that κ has a given fixed average κ0.
In the one-dimensional case our results imply in particular that for F(κ)=κ2 the constant potential fails to minimize the principal eigenvalue for α>αc=μ1/(4κ0 2), where μ1 is the first nonzero eigenvalue of −Δ. This complements a result by Exner, Harrell and Loss, showing that the critical value where the constant potential stops being a minimizer for a class of Schrödinger operators penalized by curvature is given by α c . Furthermore, we show that the value of μ1/4 remains the infimum for all α >α c . Using these results, we obtain a sharp lower bound for the principal eigenvalue for a general potential.
In higher dimensions we prove a (weak) local version of these results for a general class of potentials F(κ), and then show that globally the infimum for the first and also for higher eigenvalues is actually given by the corresponding eigenvalues of the Laplace–Beltrami operator and is never attained.
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Received: 17 July 2000 / Accepted: 11 October 2000
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Freitas, P. On Minimal Eigenvalues¶of Schrödinger Operators on Manifolds. Commun. Math. Phys. 217, 375–382 (2001). https://doi.org/10.1007/s002200100365
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DOI: https://doi.org/10.1007/s002200100365